Abstract

We investigate the ill-posed problem of minimizing weakly lower semicontinuous functionals on a convex closed set in a Hilbert space. The functionals to be minimized are available with errors. We prove that a necessary condition for the existence of a regularization procedure with a uniform accuracy estimate on the class of weakly lower semicontinuous functionals is the well-posedness of related optimization problems. We also study regularization methods that provide uniform accuracy estimates, linear with respect to the error level in cost functionals. Under appropriate assumptions on the feasible set, we establish a necessary condition for the existence of such regularizing procedures. In the case of unconstrained problems, the obtained condition reduces to the local strong convexity of functionals under minimization.

摘要

我们研究在希尔伯特空间中的凸封闭集上最小化弱较低半连续函数的不适定问题。有待减少的功能可能会出错。我们证明，对于弱下半连续函数类而言，具有一致准确度估计的正则化过程的存在的必要条件是相关优化问题的适定性。我们还研究正则化方法，这些方法可提供统一的精度估计，相对于成本函数中的误差水平，线性估计是线性的。在可行集的适当假设下，我们为此类正则化程序的存在建立了必要条件。在无约束的问题的情况下，所获得的条件在最小化的情况下减少到泛函的局部强凸性。